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In fact Kadabra does appear in the anime.

Major appearances
Sabrina's Kadabra
Sabrina, the Gym Leader of Saffron City, has a Kadabra that evolved from an Abra in Abra and the Psychic Showdown and was used against Ash. After being defeated by Sabrina, and obtaining a Haunter, Kadabra was defeated in a rematch in Haunter Versus Kadabra. It was seen again in a flashback in Bad to the Bone.
Minor appearances
A Kadabra appeared in The Mandarin Island Miss Match. It was defeated in a battle by Prima's Cloyster.
A Kadabra was seen in a fantasy in The Rivalry Revival.
In Fear Factor Phony, a Kadabra was living in an abandoned mining colony with several other Psychic-type Pokémon.

Well actually Kadabra IS in the anime. And also apparently a lawsuit against Nintendo was not the only issue. He was sued by another company and then he countersued and so on and so forth. You know how these stupid law things go :P

Kadabra is in the anime.
He appears in these shows:Sabrina's Kadabra
The Mandarin Island Miss Match
The Rivalry Revival
Fear Factor Phony

His main show though, was Sabrina's Kadabra.

Sabrina, the Gym Leader of Saffron City, has a Kadabra that evolved from an Abra in Abra and the Psychic Showdown and was used against Ash. After being defeated by Sabrina, and obtaining a Haunter, Kadabra was defeated in a rematch in Haunter Versus Kadabra. It was seen again in a flashback in Bad to the Bone. -Bulbapedia